LEADER 07637nam 22006853 450 001 9910915795003321 005 20240723121405.0 010 $a1-4704-7539-1 035 $a(MiAaPQ)EBC30671910 035 $a(Au-PeEL)EBL30671910 035 $a(PPN)272106429 035 $a(CKB)27902413000041 035 $a(EXLCZ)9927902413000041 100 $a20230804d2023 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aEigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2023. 210 4$dİ2023. 215 $a1 online resource (186 pages) 225 1 $aMemoirs of the American Mathematical Society Series ;$vv.287 311 08$aPrint version: Bruggeman, Roelof Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume Providence : American Mathematical Society,c2023 9781470465452 327 $aCover -- Title page -- Chapter 1. Introduction -- Motivational background -- Aim of this monograph -- Acknowledgement -- Part 1. Preliminaries, properties of period functions, and some insights -- Chapter 2. Notations -- Chapter 3. Elements from hyperbolic geometry -- 3.1. Models and isometries -- 3.2. Classification of isometries -- 3.3. Cusps, funnels, limit set, and ordinary points -- 3.4. Geodesics, resonances, and the Selberg zeta function -- 3.5. Intervals and rounded neighborhoods -- Chapter 4. Hecke triangle groups with infinite covolume -- Chapter 5. Automorphic forms -- 5.1. Funnel forms of different types -- 5.2. Fourier expansion -- Chapter 6. Principal series -- 6.1. Regularity at infinity -- 6.2. Presheaves and sheaves -- 6.3. Holomorphic extensions -- Chapter 7. Transfer operators and period functions -- 7.1. Discretizations and transfer operators -- 7.2. Slow transfer operators -- 7.3. Period functions -- 7.4. Real and complex period functions -- 7.5. Fast transfer operators -- 7.6. One-sided averages -- 7.7. Convergence and meromorphic extension of fast transfer operators -- 7.8. Spaces of complex period functions -- Chapter 8. An intuition and some insights -- Part 2. Semi-analytic cohomology -- Chapter 9. Abstract cohomology spaces -- 9.1. Standard group cohomology -- 9.2. Cohomology on an invariant set -- 9.3. Relation to parabolic cohomology spaces -- Chapter 10. Modules -- 10.1. Modules of semi-analytic functions -- 10.2. Submodules of semi-analytic vectors -- 10.3. Conditions on cocycles -- 10.4. Cohomological interpretation of the singularity condition -- Part 3. Automorphic forms and cohomology -- Chapter 11. Invariant eigenfunctions via a group cohomology -- Chapter 12. Tesselation cohomology -- 12.1. Choice of a tesselation, and cohomology -- 12.2. Relation to group cohomology -- 12.3. Mixed cohomology spaces. 327 $aChapter 13. Extension of cocycles -- Chapter 14. Surjectivity I: Boundary germs -- 14.1. Analytic boundary germs and semi-analytic modules -- 14.2. Cohomology classes attached to funnel forms -- 14.3. Representatives of boundary germs -- Chapter 15. Surjectivity II: From cocycles to funnel forms -- 15.1. From a cocycle to an invariant eigenfunction -- 15.2. A cocycle on an orbit of ordinary points -- 15.3. Isomorphisms -- Chapter 16. Relation between cohomology spaces -- Chapter 17. Proof of Theorem D -- From funnel forms to cocycle classes on the invariant set -- From cocycle classes on to funnel forms -- Proof of Theorem D -- Part 4. Transfer operators and cohomology -- Chapter 18. The map from functions to cocycles -- Chapter 19. Real period functions and semi-analytic cocycles -- Chapter 20. Complex period functions and semi-analytic cohomology -- Chapter 21. Proof of Theorem E -- Part 5. Proofs of Theorems A and B, and a recapitulation -- Part 6. Parity -- Chapter 22. The triangle group in the projective general linear group -- 22.1. Two actions of the projective general linear group -- 22.2. The triangle group -- Chapter 23. Odd and even funnel forms, cocycles, and period functions -- 23.1. Odd and even funnel forms -- 23.2. Odd and even cocycles -- 23.3. Odd and even period functions -- Chapter 24. Isomorphisms with parity -- Part 7. Complements and outlook -- Chapter 25. Fredholm determinant of the fast transfer operator -- Chapter 26. Outlook -- Bibliography -- Index of terminology -- List of notations -- Back Cover. 330 $a"We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society Series 606 $aAutomorphic forms 606 $aTransfer operators 606 $aHecke algebras 606 $aNumber theory -- Discontinuous groups and automorphic forms -- Automorphic forms, one variable$2msc 606 $aNumber theory -- Discontinuous groups and automorphic forms -- Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols$2msc 606 $aDynamical systems and ergodic theory -- Smooth dynamical systems: general theory -- Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems$2msc 606 $aNumber theory -- Discontinuous groups and automorphic forms -- Spectral theory; Selberg trace formula$2msc 606 $aFunctions of a complex variable -- Riemann surfaces -- Fuchsian groups and automorphic functions$2msc 606 $aDynamical systems and ergodic theory -- Dynamical systems with hyperbolic behavior -- Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)$2msc 615 0$aAutomorphic forms. 615 0$aTransfer operators. 615 0$aHecke algebras. 615 7$aNumber theory -- Discontinuous groups and automorphic forms -- Automorphic forms, one variable. 615 7$aNumber theory -- Discontinuous groups and automorphic forms -- Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols. 615 7$aDynamical systems and ergodic theory -- Smooth dynamical systems: general theory -- Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 615 7$aNumber theory -- Discontinuous groups and automorphic forms -- Spectral theory; Selberg trace formula. 615 7$aFunctions of a complex variable -- Riemann surfaces -- Fuchsian groups and automorphic functions. 615 7$aDynamical systems and ergodic theory -- Dynamical systems with hyperbolic behavior -- Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.). 676 $a512.7 676 $a512.7 686 $a11F12$a11F67$a37C30$a11F72$a30F35$a37D40$2msc 700 $aBruggeman$b Roelof$056659 701 $aPohl$b Anke Dorothea$01779803 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910915795003321 996 $aEigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume$94303362 997 $aUNINA